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Source code and simulation results for the computation of eigenfrequency sensitivities using Riesz projections for efficient optimization of nanophotonic resonators

<p><strong>Summary</strong></p> <p>Data and source code relate to the article &quot;Computation of eigenfrequency sensitivities using Riesz projections for<br> efficient optimization of nanophotonic resonators&quot; [<a href="https://doi.org/10.1038/s42005-022-00977-1">1</a>]. It combines direct differentiation of scattering problems with a contour integral method [<a href="https://doi.org/10.1016/j.jcp.2020.109678">2</a>]&nbsp;to compute eigenfrequency sensitivities. An optimization is used to demonstrate the relevance of the method.</p> <p><strong>Structure</strong></p> <p>The most important elements of this publication are the MATLAB scripts &#39;sensitivities.m&#39; and &#39;optimization.m&#39;, which can be used to reproduce the most important results of the paper. The directories&nbsp;<strong>code</strong>,&nbsp;<strong>scattering</strong>&nbsp;and&nbsp;<strong>results&nbsp;</strong>contain the software RPExpand&nbsp;[<a href="https://doi.org/10.1016/j.softx.2021.100763">3</a>], input files for JCMsuite [<a href="https://doi.org/10.1002/pssb.200743192">4</a>] and results produced with the scripts, respectively. Furthermore, the latter contains the subfolder&nbsp;<strong>tabulated,</strong>&nbsp;which contains text files&nbsp;tabulating&nbsp;data presented&nbsp;in Figures 2 and 4 of the paper. Eventually, the function &#39;code/observation.m&#39; evaluates the target for the optimization.</p> <p><strong>Additional Information</strong></p> <p>The applicaton is based on an example from the literature [<a href="https://doi.org/10.1126/science.aaz3985">5</a>]. Using apriori knowledge about the eigenmode of interest, we chose the scalar observable, as defined in Section B of the paper, to be&nbsp;the component of the electric field normal to the plane defining the solid&nbsp;of revolution.</p> <p>The convergence studies are based on the discrete, circular contour&nbsp;<span>\(\tilde{C} = \big\{ c_n~|~ c_n=r_0 e^{2\pi i n/8}, n \in \{0,1,...,7\}\big\}\)</span>&nbsp;with center <span>\(\omega_0 = 2 \pi c/(1600~\mathrm{nm})\)</span>&nbsp;and radius <span>\(r_0 = \omega_0\times10^{-2}\)</span>. For finite element degrees <span>\(d\)</span> higher than 5, the error saturates. For this reason, the differences between results for <span>\(d=5\)</span> and <span>\(d = 6\)</span> may depend on the hardware architecture.</p> <p>A larger radius&nbsp;<span>\(r = 4\times10^{13}\)</span> has been chosen for the optimization to include information from poles located further away from the frequency of interest. The target function <span>\(t(p_1,\dots,p_5) = -q_n \left(1 - \frac{(\omega_n-\omega_0)^2}{r^2} \right)\)</span>is minimized. The first factor is the negative <em>Q-</em>Factor and the second factor ensures that the target is zero at the boundary. If no eigenfrequency&nbsp;<span>\(\omega_n\)</span>&nbsp;is located inside the contour, the target is set to zero. For the purpose of this data publication some numerical parameters have been improved. This resulted in a faster convergence of the optimization.</p> <p><strong>Requirements</strong></p> <ul> <li>JCMsuite (version 5.2.0 or newer)</li> <li>MATLAB (tested with version R2019b)</li> </ul> <p>In order to run the scripts you must replace the corresponding place holders in the files by&nbsp;a path to your installation of JCMsuite. Free trial licenses are available, please refer to the homepage of&nbsp;<a href="https://jcmwave.com/">JCMwave</a>.&nbsp;</p> <p><strong>References</strong></p> <p>[1] Felix Binkowski, Fridtjof Betz, Martin Hammerschmidt, Philipp-Immanuel Schneider, Lin Zschiedrich, Sven Burger,&nbsp;Computation of eigenfrequency sensitivities using Riesz projections for efficient optimization of nanophotonic resonators, Communications Physics&nbsp;<strong>5</strong>, 202&nbsp;(2022),&nbsp;https://doi.org/10.1038/s42005-022-00977-1</p> <p>[2] Felix Binkowski, Lin Zschiedrich,&nbsp;Sven Burger,&nbsp;A Riesz-projection-based method for nonlinear eigenvalue problems,&nbsp;Journal of Computational Physics&nbsp;<strong>419</strong>, 109678 (2020),&nbsp;https://doi.org/10.1016/j.jcp.2020.109678</p> <p>[3]&nbsp;Fridtjof Betz, Felix Binkowski, Sven Burger, RPExpand: Software for Riesz projection expansion of resonance phenomena, SoftwareX <strong>15</strong>,&nbsp;100763 (2021), https://doi.org/10.1016/j.softx.2021.100763</p> <p>[4] Jan Pomplun, Sven Burger, Lin Zschiedrich, Frank Schmidt,&nbsp;Adaptive finite element method for simulation of optical nano structures, Physica Status Solidi B&nbsp;<strong>244</strong>, 3419 (2007),&nbsp;http://dx.doi.org/10.1002/pssb.200743192</p> <p>[5]&nbsp;Kirill Koshelev, Sergey Kruk, Elizaveta Melik-Gaykazyan, Jae-Hyuck Choi, Andrey Bogdanov, Hong-Gyu Park, Yuri Kivshar,&nbsp;Subwavelength dielectric resonators for nonlinear nanophotonics, Science&nbsp;<strong>367</strong>, 288 (2020), http://dx.doi.org/%2010.1126/science.aaz3985</p>